Purely affine elementary su(N) fusions

TL;DR

This paper demonstrates that for every su(N>3), there exists at least one purely affine elementary fusion, which is a fundamental fusion not reducible to elementary couplings, extending previous observations in su(4).

Contribution

It constructs explicitly the existence of purely affine elementary fusions for all su(N>3), highlighting a new fundamental aspect of affine fusion structures.

Findings

Existence of purely affine elementary fusions for all su(N>3)

Construction method for these fusions

Extension of previous su(4) observations

Abstract

We consider three-point couplings in simple Lie algebras -- singlets in triple tensor products of their integrable highest weight representations. A coupling can be expressed as a linear combination of products of finitely many elementary couplings. This carries over to affine fusion, the fusion of Wess-Zumino-Witten conformal field theories, where the expressions are in terms of elementary fusions. In the case of su(4) it has been observed that there is a purely affine elementary fusion, i.e., an elementary fusion that is not an elementary coupling. In this note we show by construction that there is at least one purely affine elementary fusion associated to every su(N>3).